Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations

This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.